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In 1991 Tratnik derived two systems of multivariable orthogonal Racah polynomials and considered their limit cases. q-Extensions of these systems are derived, yielding systems of multivariable orthogonal q-Racah polynomials, from which systems of multivariable orthogonal q-Hahn, dual q-Hahn, q-Krawtchouk, q-Meixner, and q-Charlier polynomials follow as special or limit cases.

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Let Pn(α,β) be the Jacobi polynomial of degree n, order (α,β), α,β > – 1, defined by[9, p. 67], and let Rn(α,β)(x) = Pn(α,β)(x)/Pn(αβ)(1). Then for n ≧ m,whereSince Rn(α, β)(l) = 1, it follows that(1)It is known that if (the ultraspherical case) or if α = β + 1, then α = β + 1, then g(k, n, m) ≧ 0.

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In 1991 Tratnik derived two systems of multivariable orthogonal Wilson polynomials and considered their limit cases. q-Analogues of these systems are derived, yielding systems of multivariable orthogonal Askey-Wilson polynomials and their special and limit cases.

Abstract. An indefinite bibasic sum containing three parameters is evaluated and used to derive bibasic extensions of Euler's transformation formula and of a Fields and Wimp expansion formula. It is also used to derive a transformation formula involving four independent bases, a 9-Lagrange inversion formula, and some quadratic, cubic and quartic summation formulas.